TY - JOUR

T1 - Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains

AU - Poláčik, P.

AU - Terracini, Susanna

PY - 2014/2/10

Y1 - 2014/2/10

N2 - We consider a semilinear elliptic equation on a smooth bounded domain Ω in ℝ2, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly positive, they are also decreasing in x for x > 0. Our goal is to exhibit examples of equations which admit nonnegative, nonzero solutions for which the second property fails; necessarily, such solutions have a nontrivial nodal set in Ω. Previously, such examples were known for nonsmooth domains only.

AB - We consider a semilinear elliptic equation on a smooth bounded domain Ω in ℝ2, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly positive, they are also decreasing in x for x > 0. Our goal is to exhibit examples of equations which admit nonnegative, nonzero solutions for which the second property fails; necessarily, such solutions have a nontrivial nodal set in Ω. Previously, such examples were known for nonsmooth domains only.

KW - Nodal set

KW - Nonnegative solutions

KW - Planar domain

KW - Semilinear elliptic equation

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U2 - 10.1090/S0002-9939-2014-11942-3

DO - 10.1090/S0002-9939-2014-11942-3

M3 - Article

AN - SCOPUS:84893306423

VL - 142

SP - 1249

EP - 1259

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 4

ER -